Place cauliflower florets in a food processor; pulse into rice-sized pieces.
Heat butter and oil in a large skillet over medium heat. Add onion; cook and stir for 2 minutes. Add cauliflower "rice" and mushrooms; cook, stirring frequently, until softened, about 5 minutes. Stir in parsley, garlic, and poultry seasoning. Cook until fragrant, about 1 minute. Remove from heat.
Stir chicken, cream of mushroom soup, mozzarella cheese, milk, salt, and pepper into the "rice" mixture.
Stand bell peppers cut-side up in a baking dish. Stuff bell peppers with chicken mixture, pushing it down to fill them completely. Sprinkle with cayenne pepper. Cover baking dish with aluminum foil.
Bake in the preheated oven until heated through, about 30 minutes. Uncover and continue baking until golden brown, about 15 minutes more.